Lindahl Equilibrium

Lindahl Equilibrium and Personal Prices for Public Goods in Health

A Lindahl equilibrium is a kind of equilibrium for an economy with a public good, made up of one quantity of the good and a set of cost shares at which every person, paying their own share of each unit, asks for exactly that quantity. Prices differ between people while the quantity is the same for all. Health economists meet the idea because many health benefits are shared, such as lower infection risk in a population or the knowledge produced by drug research, and the Lindahl outcome describes efficient provision when each beneficiary pays in line with their marginal willingness to pay. The same model shows why that outcome is hard to reach, since people gain by hiding what the good is worth to them.

Why one price fails for a shared good

The Stanford Encyclopedia of Philosophy entry on public goods defines them as both non-rivalrous and non-excludable, and lists public health as an example, because the fewer people infected with a contagious disease, the less likely any healthy person is to catch it. HM Treasury's Green Book (2026) lists the failure to provide public goods among market failures, stating that the free market under-provides them.

For a private good every buyer faces the same price and chooses a different quantity, so each person's marginal rate of substitution ends up equal to that common price. For a public good everyone consumes the same quantity, so a single price cannot match each person's valuation. Efficient provision instead needs the condition usually credited to Samuelson's 1954 paper.

$$\sum_{i=1}^{N} MRS^{i}_{G,x} = MRT_{G,x}$$

where $N$ is the number of people sharing the good, $MRS^{i}_{G,x}$ is the amount of the private good $x$ that person $i$ would give up for one more unit of the public good $G$ (their marginal benefit), and $MRT_{G,x}$ is the amount of the private good that must be given up to produce one more unit of $G$, its marginal cost. Luca Spataro's public economics notes explain that an extra unit raises every household's utility, so its social benefit is the sum of the marginal benefits.

Personal prices and the equilibrium conditions

Spataro's notes argue that because consumers differ in the value they place on a given supply, one price for all cannot be efficient, whereas a personalised price reflecting each consumer's valuation can be. Such prices extend the Arrow and Debreu model, in which each commodity had a single price, and the resulting equilibrium is named after Lindahl's 1919 work. Lise Vesterlund's Pittsburgh lecture on mechanism design defines it as a set of cost shares summing to one and a provision level that every person demands at their own share.

$$\sum_{i=1}^{N} t_i = 1, \qquad G^{*} = G_1(t_1) = G_2(t_2) = \dots = G_N(t_N), \qquad MRS^{i}_{G,x} = t_i \, c$$

where $t_i$ is person $i$'s cost share, $c$ is the constant cost of one unit of the public good, $t_i c$ is person $i$'s personal (Lindahl) price, $G_i(t_i)$ is the quantity person $i$ would choose when paying that price for every unit, and $G^{*}$ is the equilibrium quantity. Adding the last condition across people gives a sum of marginal rates of substitution equal to $c$, the Samuelson condition, so the equilibrium quantity is efficient.

Roemer and Silvestre trace the idea to a parliament, in which Wicksell in 1896 and Lindahl in 1919 pictured parties negotiating until no universally beneficial change was left, so the unanimous agreement is Pareto efficient. Lindahl's thesis, written under Wicksell, modelled two parties, one for the rich and one for the poor. A person's cost share at such an equilibrium is their Lindahl ratio, their marginal valuation of the public good divided by the social marginal valuation.

Efficiency, the core and voluntary contributions

The Lindahl allocation reaches Pareto efficiency through something like market behaviour. Spataro's notes call this a form of the First Theorem of Welfare Economics, add that redistributing the initial endowment gives a new Lindahl equilibrium at another efficient point, and state that the equilibrium is in the core. Roemer and Silvestre describe the core as a condition of social stability under which no group can improve on the outcome.

The usual comparison is with voluntary provision, which is in general not efficient. Roemer and Silvestre treat it as a game in which each person chooses a contribution, analysed with the Cournot and Nash equilibrium concept, whose typical inefficiency is the free rider problem. Buchholz, Cornes and Peters start from the known result that a Lindahl equilibrium is not necessarily Pareto superior to the Cournot and Nash outcome, and show that everyone gains when the income distribution is not too skewed or the number of countries is large, and apply this to international cooperation on global public goods, the setting of the drug research case below.

Worked example with a regional outbreak surveillance service

All numbers in this example are illustrative. Three local areas, A, B and C, are deciding how many staff-months a year of a shared infection surveillance and outbreak response service to fund. None can be shut out of the warnings, and each gains from earlier detection wherever an outbreak starts. Marginal willingness to pay, in GBP thousand per staff-month, is $v_A = 80 - G$, $v_B = 50 - G$ and $v_C = 35 - G$, where $G$ is the number of staff-months, and each staff-month costs GBP 75 thousand.

The Samuelson condition sets the sum of marginal benefits equal to the marginal cost.

$$(80 - G) + (50 - G) + (35 - G) = 165 - 3G = 75$$

where the left side is the combined marginal willingness to pay at quantity $G$ and 75 is the cost of a staff-month in GBP thousand. Solving gives $G^{*} = (165 - 75)/3 = 30$ staff-months.

The Lindahl prices are each area's marginal benefit at 30 staff-months, so $p_A = 80 - 30 = 50$, $p_B = 50 - 30 = 20$ and $p_C = 35 - 30 = 5$, in GBP thousand per staff-month. They add up to the unit cost, $p_A + p_B + p_C = 50 + 20 + 5 = 75$, so the cost shares are $t_A = 50/75 \approx 0.67$, $t_B = 20/75 \approx 0.27$ and $t_C = 5/75 \approx 0.07$. At those prices each area, choosing alone, asks for 30 staff-months. Payments are $P_A = 50 \times 30 = 1500$, $P_B = 20 \times 30 = 600$ and $P_C = 5 \times 30 = 150$, in GBP thousand, totalling $P = 1500 + 600 + 150 = 2250$, the full cost of $K = 75 \times 30 = 2250$.

Each area's net gain is the area under its marginal benefit line up to 30 staff-months minus its payment. That gives $S_A = 80 \times 30 - 450 - 1500 = 450$, $S_B = 50 \times 30 - 450 - 600 = 450$ and $S_C = 35 \times 30 - 450 - 150 = 450$, where $S$ is net gain and $\Delta = 30^2/2 = 450$ is the same triangle for each area. The equal gains come from the identical slopes chosen here, not from any general property.

Under voluntary contributions, A funds the service until $v_A = 80 - G = 75$, which gives only 5 staff-months at a cost to A of $P_A = 75 \times 5 = 375$. At that level B and C value one more staff-month at $v_B = 50 - 5 = 45$ and $v_C = 35 - 5 = 30$, below the cost of 75, so they contribute nothing. With equal shares, each area pays $p = 75/3 = 25$ per staff-month and asks for 55, 25 and 10 staff-months, so a majority vote settles on B's 25.

Net gains are in GBP thousand a year. Voluntary funding leaves most of the possible gain unrealised, with B and C free riding on A. Equal shares come close on the total but leave C worse off than without the service, paying $P_C = 25 \times 25 = 625$ for benefits worth $V_C = 35 \times 25 - 312.5 = 562.5$. Only the Lindahl prices reach the efficient quantity with every area gaining, because they tie the quantity decision to the financing decision.

Use in cost-benefit analysis and health policy

The Samuelson condition is the quantity rule behind cost-benefit analysis of a shared programme, which sums willingness to pay across the people affected and compares it with cost. Brouwer and colleagues describe applied cost-benefit analysis as potential Pareto improving, since each gain or loss carries the same weight whoever bears it. The Lindahl equilibrium is the case in which the potential becomes actual, because a core allocation leaves no person or group able to do better alone, so the programme is a Pareto improvement compared with providing nothing.

Frech, Pauly, Comanor and Martinez treat pharmaceutical research and development as a global public good. In their Lindahl model each country pays its own valuation at the optimal quantity, satisfying the Samuelson condition. They argue that each government is strongly motivated to understate its valuation to cut its contribution, most of all in small countries, and that with no higher authority to enforce Lindahl behaviour the good may be under-provided. Lindahl quantities, in their view, are a useful standard but not incentive-compatible.

The same problem limits survey valuation of shared health goods. The Stanford entry notes that Lindahl taxes would give an efficient equilibrium if preferences were observable, but that people have an incentive to understate their reservation prices and free ride. A contingent valuation survey of a public health programme faces this risk when respondents think their answers will set their payment.

Welfarist assumptions and the extra-welfarist objection

The Lindahl equilibrium belongs to welfare economics in its welfarist form. Each price is a person's own marginal valuation, and Vesterlund's lecture notes that when the good is normal and preferences are alike, the Lindahl price rises with wealth, because richer people want more of the good. Brouwer and colleagues describe Paretian economics as treating individual utilities as the outcome, holding them interpersonally incomparable and taking the initial income distribution as given, and a Lindahl allocation shares all three features.

Roemer and Silvestre note that Lindahl's rule yields a marginal version of the benefit principle of taxation, under which taxes follow the benefit received from public goods. They do not accept that marginal cost pricing is associated with distributive justice, beyond efficiency being one aim of justice.

Extra-welfarist health economics allocates care by need rather than by willingness to pay. Brouwer and colleagues describe cost-utility analysis as using something like need in place of effective demand and willingness to pay, a concept unrelated to ability to pay. They quote the 1944 white paper A National Health Service, which promised that getting care would not depend on whether people could pay for it or on any other factor irrelevant to real need. A tax-financed service that allocates care by need therefore departs from Lindahl pricing by design, although the Lindahl quantity remains a test of whether a shared good such as surveillance is supplied in the efficient amount.

Neighbouring ideas and where each applies

Club goods are shared without rivalry but are excludable, so a membership charge can finance them without personal prices. A Pigouvian tax corrects an externality by pricing the harm, while a Lindahl price finances a good everyone consumes, although Roemer and Silvestre note the two problems are hard to separate sharply. Herd protection from immunisation is usually framed as a positive externality, the approach of the vaccination externalities page. Mechanism design builds rules, such as the Groves and Clarke mechanism in the Stanford entry, under which people have no incentive to misstate their valuations, which is what an incentive-compatible mechanism means.

Common misreadings of Lindahl equilibrium

Most errors treat the equilibrium as a working funding formula or as a verdict on fairness. Three recur in health applications.

1. Treating it as a self-running mechanism. Vesterlund's lecture stresses that no market process generates the Lindahl prices and that people gain by underreporting demand.
2. Reading efficiency as fairness. The prices follow marginal valuations, which depend on income, so a Lindahl allocation can be efficient and still unacceptable on equity grounds.
3. Confusing the quantity rule with the financing rule. The Samuelson condition fixes how much to provide, and the Lindahl prices add who pays.

Sources

• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003
• Buchholz W, Cornes R, Peters W. Lindahl vs. Cournot Nash: the role of the income distribution. University of Nottingham GEP Research Paper 2004/27. 2004. https://nottingham.ac.uk/gep/documents/papers/2004/04-27.pdf
• Frech HE III, Pauly MV, Comanor WS, Martinez JR. Pharmaceutical pricing and R&D as a global public good. NBER Working Paper 31272. May 2023. https://www.nber.org/papers/w31272
• HM Treasury. The Green Book (2026). Updated 28 October 2026. Chapter on rationale and objectives (market failures). https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• Reiss J. Public goods. In: Zalta EN, ed. The Stanford Encyclopedia of Philosophy. First published 21 July 2021. https://plato.stanford.edu/entries/public-goods/
• Roemer JE, Silvestre J. Kant and Lindahl. Cowles Foundation Discussion Paper 2278, Yale University. February 2021. https://cowles.yale.edu/sites/default/files/2022-08/d2278_0.pdf
• Spataro L. Public economics. Market failures, public goods, first part. Lecture slides, University of Pisa. 2015. https://publicecoeu.ec.unipi.it/wp-content/uploads/2015/03/3_Public-goods_Part_One.pdf
• Vesterlund L. Econ 2230 Public Economics, Lecture 8, Mechanism design. Lecture slides, University of Pittsburgh. 2011. https://sites.pitt.edu/~vester/2230_l8.pdf